Finite difference simulation of coarse grained PDEs to describe 2D transport limited patterns in autocatalytic reactions. Bifurcation and perturbation theoretic methods to determine parameter ranges for metastable patterns.

1. 1
Formation and Stability of mixing-limited Patterns in Homogeneous
Autocatalytic Reactors
Thesis submitted in partial fulfillment of the
requirements for the degree
of
Master of Technology
In
Chemical Engineering
By
Tanmoy Sanyal
Roll No. 08CH3025
UNDER THE SUPERVISION
OF
Dr. Saikat Chakraborty
DEPARTMENT OF CHEMICAL ENGINEERING
INDIAN INSTITUTE OF TECHNOLOGY, KHARAGPUR
2013

2. .
2
Department of Chemical Engineering,
Indian Institute of Technology, Kharagpur-721302.
________________________________________________________________________
CERTIFICATE
This is to certify that the thesis entitled “Formation and Stability of mixing-limited
Patterns in Homogeneous Autocatalytic Reactors” submitted by Mr. Tanmoy
Sanyal, to the Department of Chemical Engineering, in partial fulfillment for the award
of the degree of Bachelor of Technology is an authentic record of the work carried out by
him under my supervision and guidance. The thesis has fulfilled all the requirements as
per the regulations of this institute and, in my opinion, has reached the standard needed
for submission.
Date: 3rd
May, 2013 ---------------------------------
Dr. Saikat Chakraborty
Department of Chemical Engineering
Indian Institute of Technology, Kharagpur

3. 3
ACKNOWLEDGEMENT
This project work would not have been possible without the guidance of Dr. Saikat Chakraborty.
I express my sheer indebtedness to him for the suggestions and stimulating discussions
throughout the year that were vital for shaping this project and for his cordial treatment and
consistent moral support.
I am thankful to all my professors in the department who have been a source of
inspiration. I express my sincere gratitude towards them for helping me reach the state that I am
today.
Finally, I wish to express my heartfelt gratitude to my friends and classmates, staff
members, and others who directly or indirectly helped me in carrying out this project.
Tanmoy Sanyal
Roll No. 08CH3025
Department of Chemical Engineering
Indian Institute of Technology, Kharagpur

4. 4
Abstract
List of Figures and Tables
Nomenclature
CHAPTER 1: Introduction 9
1.1 Pattern formation in nature 10
1.2 Transport limited pattern formation in autocatalytic reactions 10
1.3 Stability of patterns 10
1.4 Objective and Organization of this thesis 11
CHAPTER 2: Literature Review 12
2.1 Mathematical modeling of pattern formation 13
2.2 A brief description of bifurcation theory (as relevant to this project) 14
CHAPTER 3: Formulation of Mathematical Model 16
3.1 Three dimensional complete model 17
3.2 Spatial averaging and regularization of model 19
CHAPTER 4: Steady State Analysis 21
4.1 Bifurcation diagrams 22
4.2 Hysteresis locii 23
4.3 Linear stability analysis 24
4.4 The Arc Length Method 27
CHAPTER 5: Dynamic Simulation of Patterned States 29
5.1 Simulation algorithm 30
5.2 Results and discussion 31
CHAPTER 6: Stability of Patterned States 36
6.1 The notion of stability 37
6.2 Approaches to stability analysis of dynamic states 37
6.3 Frechet Derivatives and their computation 39
6.4 The 2-D Finite Fourier Transform in plane polar co-ordinates 40
6.5 Stability maps and their time development for a few test-cases 41
CHAPTER 7: Conclusions and Future Work 51
CHAPTER 8: References 53

5. 5
Abstract
Transport limited patterns have always been a very important area of analysis considering its application
to the industry. When a steady state concentration in a tubular reactor system loses its stability to
transverse as well as axial perturbations, it may give rise to patterned states, resulting in difference in
concentration and temperatures throughout the geometry. The study of such dynamic states is important is
necessary in order to understand how to eliminate such pattern formation, since they are responsible for
product degradation and thermal runaways in reactors. In this project, we study the formation and
stabilities of concentration patterns in an isothermal system resulting out of a fast homogenous
autocatalytic reaction of the form A+2B 3B. We do not work with the complete 3-D model but apply
the Liapunov Schmidt scheme of classical Bifurcation Theory to construct low dimensional models by
averaging over the axial dimension. The resulting models are simpler, computationally less formidable
and at the same time, preserve all relevant information about the axial dimension by retaining the axial
Peclet number in the final governing equations. Steady state bifurcation analysis is carried out on this
system to determine the regime in the parameter space (Damkohler no., transverse and axial Peclet nos.)
favorable to pattern formation. Once the regime has been determined, proper operating parameters are
selected and computer simulations are performed to capture the spatio-temporal evolutions of the pattered
states, starting with a uniform base state. The effect of different parameters on the time it takes for a
pattern to appear and wash out into uniformity is studied. Finally, we define the concept of stability for
dynamic states and outline the various approaches possible for studying such stability. We propose to do a
bifurcation analysis of the patterned states themselves, and compute stability maps which show how
stability and instability zones vary across the pattern with passing time.

6. 6
NOMENCLATURE
a = radius of the tubular reactor
L = length of the reactor
c=dimensionless reactor concentration
= dimensionless initial reactor concentration
= axially averaged dimensionless reactor concentration
= dimensionless exit concentration
= dimensionless steady state concentration
= dimensionless perturbation concentration term
C = reactant concentration
= inlet reactor concentration
= initial concentration in the reactor
Da = Damkohler number
D = molecular diffusivity
f = Eigen function in axial direction
= Bessel function of the first kind
k = reaction rate constant
m = azimuthal mode number
n = radial mode number
p = transverse Pe´clet number
Pe = axial Pe´clet number
r = dimensionless reaction rate, radius variable
R(A)/(B ) = rate of disappearance of A or B
t =dimensionless time
t′ =time
= characteristic radial diffusion time

7. 7
= characteristic axial diffusion time
characteristic reaction time
u'(x) = velocity field
u(x) = dimensionless velocity field
X(A)/(B ) = conversion of species A or B
z = dimensionless axial spatial coordinate
= azimuthal spatial coordinate
coefficient of time in the exponential time
= nth
eigenvalue of the mth
mode
= inlet feeding ratio of B to A
= dimensionless radial coordinate
= 2-D fourier transform in plane polar co-ordinates of and
= 2-D finite fourier transform in plane polar co-ordinates
= particular patterned state (time snapshot)/
F = state vector of governing equations of patterned states.
= Frehet Derivative operator about the point
= Frechet derivative of about the point .

8. 8
LIST OF FIGURES AND TABLES
FIGURE NO. DESCRIPTION
1 Phase portrait showing saddle node bifurcation
2 One and Two Limit point bifurcations
3 Tubular Reactor
4 Bifurcation diagrams with parameter Pe
5 Bifurcation diagrams with parameter
6 Hysteresis Locii for =0.05, showing stable and unstable regions
7 Traversal of the bifurcation diagram with time
8 Neutral Stability curves for different eigenmodes
9 Neutral Stability analysis for dynamic states
10 Bifurcation analysis of dynamic state
11 Stability of modes expressed as % of stable node points
12 Stability and Segregation (conflicting trends) for m=5, n=5
TABLE NO. DESCRIPTION
1 Different values of for different m and n
2 The three basic patterns
3 Patterns for different with m=1,n=1, Pe =1, p=100
4 Patterns for different Pe with m=1,n=1, p=100
5 Patterns for different p with m=1,n=1, Pe=1
6 Stability map for , Pe=1 , p=100, Da=14
7 Stability map for , p=100 and Pe = 0,5 and 10
8 Stability Maps of symmetric eigenmodes for Pe=1, Da=10 and critical p for
the mode
9 Stable patterns for various mode numbers

9. 9
CHAPTER 1
INTRODUCTION

10. 10
INTRODUCTION
1.1 Pattern formation in nature
Interaction between transport and reaction rate processes in chemical reactors gives rise to a
variety of spatio-temporal patterns resulting from multiple steady states. Spatial pattern
formation was first studied by Turing [1] in 1952 in his seminal work titled "The Chemical Basis
of Morphogenesis". Later Nicolis and Prigogine [2] suggested a mechanism of symmetry-
breaking and pattern formation in non-equilibrium systems, with Prigogine and coworkers
developing the Brusselator model [3–5] that exhibits Turing instability.
1.2 Transport limited pattern formation in autocatalytic reactions
In this investigation, we mainly concentrate on autocatalytic reaction systems. Transport-limited
patterns are generated in such systems when a spatially uniform steady state loses its stability to
transverse perturbations, and the patterns, once formed, are sustained when the timescale of local
diffusion processes are much larger compared to the overall reaction timescale. The best example
of this phenomenon can be seen in the formation of localized zones of higher concentrations and
temperature in reactors, commonly referred to as 'hot spots'. This implies the existence of
asymmetrical temperature and concentration profiles across the cross section of a reactor.
Temperature patterns were observed in packed bed reactors during partial oxidation of isobutyl
alcohol by Boreskov et al. [6] and Matros [7], in trickle bed reactors by Barkelewand Gambhir
[8], and in radial flow and packed bed reactors by Luss and coworkers [9,10]. Such a condition
can often decrease the yield of the desired product, deactivate the catalyst and initiate 'thermal
runaways' i.e. highly exothermic undesirable reactions. All of these lead to isafety hazards and
more imporatantly decreases the reactor strength and product quality.
1.3 Stability of patterns
Speaking about the stability of a pattern essentially means speaking about the stability of a
dynamic state. Since a pattern is a spatially and temporally continually evolving and changing
state, the concept of stability for a pattern will not be the same as that for a steady uniform state.
It must be clearly understood that the formation of a pattern state is a product of instability too,
but that is an instability of the base state and is significantly different from the concept of

11. 11
instability of the patterned state itself. A transport limited pattern will always eventually wash
out due to the effect of diffusion, but before doing so, if it remains unchanged for a significant
amount of time, such a pattern may be called a stable pattern. Stable thermal patterns, are more
often than not responsible for thermal runaways in a reactor. Later we shall find, why it is
imperative to define stability in this way and not in other conventional ways.
1.4 Objective and organization of this thesis
In this investigation we study spatio-temporal pattern formation for homogenous autocatalytic
reactions of the form A+2B→3B. We start with the full 3-D Convection Diffusion Reaction
model which we reduce to a two dimensional two-mode model that retains all the parameters of
the original equation. The dimensionality reduction is done using the Liapunov-Scheme of
classical Bifurcation theory, which is in essence a spatial averaging scheme, but it retains
important information about the physics along the dimension that is smoothed out due to
averaging. Further, we extend the region of validity of this axially averaged two-mode model by
using a mathematical procedure called regularization to obtain a regularized model, which is
then subjected to the same treatment. Post spatial averaging and regularization we obtain steady
state bifurcation diagrams under different inlet conditions of the 2 species A and B, showing the
existence of multiple steady states. The stability of these multiple solutions to transverse
perturbations is examined using linear stability analysis. Analytical expressions describing steady
state solutions and neutral stability conditions are obtained for the case of low-dimensional
models, which are very difficult to obtain for the 3D model. The bifurcation diagrams and the
neutral stability maps helps us to determine the region of instability in terms of system
parameters i.e. the region that induces pattern formation. We follow this with a dynamic
simulation of patterned states using the regularized 2-D model for an isothermal case. Then we
proceed to define the concept of stability for dynamic states, analyse the two most common
methods of stability analysis and propose to use the bifurcation analysis of the patterned states.
We generate a dynamic hysteresis locus for a pattern and with it determine how regions of
stability and instability are distributed across the pattern. Such a map is termed as a stability map
and we study the temporal evolution of the stability maps of two test cases.

12. 12
CHAPTER 2
LITERATURE REVIEW

13. 13
LITERATURE REVIEW
2.1 Mathematical modeling of pattern formation
Most modeling attempts in the literature aimed at predicting pattern formation have been focused
on heterogeneous packed bed and catalytic reactors. Early theoretical studies by Luss et al. [11]
investigated the existence of asymmetric steady states in catalytic slabs using one-dimensional
diffusion-reaction models. Schmitz and Tsotsis [12] showed in their theoretical study that inter-
particle interactions give rise to spatially patterned states under certain conditions. Balakotaiah
and coworkers [13–15] have shown that flow misdistributions and hot spots may occur in down-
flow packed-bed reactors and the regions of these instabilities are determined in terms of various
transport and kinetic parameters. Benneker et al. [16] indicated that hydrodynamic instabilities
observed in packed-bed reactors may disturb the plug-flow character and may lead to hot spots
and deactivated catalysts. Balakotaiah and coworkers predicted transverse pattern formation in
adiabatic packed bed reactors in which a bimolecular reaction (with Langmuir–Hinshelwood
kinetics) occurs [17], and in catalytic monolith reactors in which an exothermic surface reaction
occurs [18]. Sheintuch and Nekhamkina [19] analyzed the pattern formation in homogeneous
model of a fixed catalytic bed for reactions with oscillatory kinetics. Mathematical models
developed to study chemical reactors are derived from fundamental balances of species, energy,
momentum, in conjunction with various constitutive relationships. The resulting model consists
of a set of unsteady state three-dimensional partial differential equations containing a large set of
physiochemical parameters. Significant complexity is introduced due to the non-linear
dependence of kinetic and transport coefficients on the state variables. The usual approaches of
modeling are either the bottom-up approach or the top-down approach. The bottom-up method
consists of rigorous computational fluid dynamics (CFD) that enables one to explore the
solutions of the three-dimensional convection-diffusion-reaction (CDR) equation in the multi-
dimensional parameter space .However, it is numerically very expensive, and is fairly
impractical even with present day computational technology, especially when incorporating the
model to existing control strategies. On the other hand, the top-down approach makes a priori
oversimplifying assumptions on the length and time scales of reaction, convection and diffusion
and then applies conservation equations only at the macroscopic levels. Thus, though easy to
solve this scheme is incapable of capturing the complex spatio-temporal reactor behaviors such

14. 14
as multiplicity, pattern/hot-zone formation, and reactor runaway that are observed during
operation. Accurate low-dimensional models that are numerically inexpensive yet retain all the
qualitative features of the 3D CDR model are required for the purpose of design, control and
optimization of a chemical process. Such an intermediate approach has been presented by
Chakraborty and Balakotaiah [20–23], in which the fundamental three-dimensional CDR
equation is averaged or homogenized over the smaller length (time) scales using Liapunov–
Schimdt (L–S) technique [24] of the classical bifurcation theory to obtain low-dimensional
models that retain all the parameters and therefore all the spatio-temporal features of the full
CDR equation. The reduced dimensionality of the models substantially reduces the
computational expense required, thus making it suitable for engineering applications.
2.2 Elements of classical Bifurcation Theory
Bifurcation theory is the mathematical study of changes in the qualitative
or topological structure of a given family, such as the integral curves of a family of vector fields,
and the solutions of a family of differential equations. Most commonly applied to
the mathematical study of dynamical systems, a bifurcation occurs when a small smooth change
made to the parameter values (the bifurcation parameters) of a system causes a sudden
'qualitative' or topological change in its behavior. Bifurcations occur in both continuous systems
(described by ODEs, DDEs or PDEs), and discrete systems (described by maps). The name
"bifurcation" was first introduced by Henri Poincaré in 1885 in the first paper in mathematics
showing such a behavior. Henri Poincaré also later named various types of stationary points and
classified them.
The key concepts of the theory as relevant to the investigation at the hand are that –
a) A nonlinear equation where x is a variable and p can always represent the
steady state form of a dynamical system given by . Since is
nonlinear it can have several different solutions. Equivalently, the dynamical system may
have several steady states depending on the parameter p, called the bifurcation parameter
popularly.
b) Our investigation would require construction of a bifurcation diagram. In dynamical
systems, a bifurcation diagram shows the possible long-term values (equilibria/fixed
points or periodic orbits) of a system as a function of a bifurcation parameter in the

15. 15
system. It is usual to represent stable solutions with a solid line and unstable solutions
with a dotted line.
Figure 1 – Phase portrait showing saddle node bifurcation
c) The central concept that we must introduce at this
stage to be used later is the traversal of a limit
point. If the bifurcation curve has only 1 limit
point, (show figure), then the steady state value
even if it occassionally climbs up onto the middle
branch will come down to the steady lower branch
called the extingushed state.
However, if there are two or more limit points then the solution when perturbed may
traverse the middle branch completely and go
beyond the second limit point into the upper branch
(show figure), called the ignited state. This means
that the number of limit points in the bifurcation
diagram of a state directly decides what the fate of
the state would be under perturbation.
Figure 2a – One Limit point
Figure 2b – Two Limit points

16. 16
CHAPTER 3
FORMULATION OF
MATHEMATICAL
MODEL

17. 17
MATHEMATICAL MODEL
3.1 Three dimensional complete model
We consider a tubular reactor of length L and hydraulic radius a in which a homogeneous
autocatalytic reaction given by
(1)
occurs, and its rate equation is given by
(2)
where k is the reaction rate constant. Our starting point is the Convection-Diffusion-Reaction
equation written in three dimensions, describing the mass balances of species A and B over the
entire domain of the reactor.
(1)
where, is the velocity field and is the effective diffusion coefficient of species i ( i
=A,B). When written out in full, along with the necessary boundary conditions, we have,
(2)
where are the three spatial dimensions involved. The relevant boundary conditions are
finite concentration at the center, Neumann conditions of zero flux at the outer walls of the tube,
Danckwerts conditions at the reactor inlet and finally periodic boundary conditions in the
azimuthal direction.
is finite (4)
(5)
(6)
(7)
Figure 3 – Tubular Reactor

18. 18
(8)
Any initial distribution of species can be taken to solve the system,
(9)
Taking L and a as characteristic dimensions in the axial and radial directions, as a reference
concentration and as average velocity, we can classify the 3-D transport into several
representative time-scales —
 Convection time scale
 Axial diffusion time scale
 Radial diffusion time scale
 Reaction time scale
Using these timescales, we obtain four independent non-dimensional groups that help
parameterize the system, viz.
 Transverse (or radial) Peclet Number,
 Axial Peclet Number,
 Damkohler Number,
As stated earlier, we are interested in studying transport limited patterns and hence assume the
molecular diffusivities to be equal (which may otherwise give rise to Turing patterns) i.e.
. Next we proceed to non-dimensionalise the governing equations in terms of the
following non-dimensional variables and groups,
to obtain,
(9)
where, .The boundary and initial conditions are given by,
and (10)
is finite and (11)

19. 19
(12)
and an appropriate initial condition given by
(13)
where, , and .
3.2 Spatially Averaged and Regularized Model
In this investigation, we consider very fast reactions. This is necessary to establish a significant
difference in time-scales of radial and axial diffusion with that of reaction, and thus ensure that
patterned states if produced, will be transport limited and not be affected majorly by the reaction
rate. What this also means is that the reaction will be completed within a very short axial length
of the reactor. This effective reactor length ( ) is given by
(14)
where the factor depends on the flow profile. For fully developed parabolic velocity profiles
in a tubular reactor, . From Eq. (14), we note that for very less , will also be
very small compared to the radius a. This hints towards doing a lumped parameter analysis by
removing the axial dimension. However, the axial Peclet Number (Pe) is an important parameter
and cannot be simply left out of the analysis. So, instead we propose to spatially average the
model along the axial dimension.
The above 3-D model is subjected to Liapunov-Schmidt averaging along the axial co-ordinate.
The details of the L-S averaging procedure for homogenous reactions in tubular geometries can
be found in [20]. Here, we directly present the spatially averaged model which captures the
spatio temporal variation along radial and azimuthal directions. L-S averaging reduces the
dimensionality of the model and results in a 2-D system involving the radial and azimuthal
dimensions. However, as discussed earlier it vital information about the axial dimension by
preserving the parameter Pe .The final model is contains not one but two independent modes –
the master mode represented by the axially averaged species concentration,
(15)
and a secondary mode defined by the outlet concentration,

20. 20
(16)
In terms of these two modes, the new model can be cast as a global evolution equation that
describes the spatio-temporal evolution of the master mode and another algebraic relation that
defines the inter-communication between the master and secondary modes and is representative
of the axial mixing in the system. In this analysis, we skip the detailed derivation and present the
final model equations.
Global Equation:
(17)
Local Equation:
(18)
It has been established [20] that the radius of the convergence of the local equation is given by
or . Outside this region, the quantitative accuracy and sometimes even the
qualitative nature of the low dimensional models are questionable. Hence, to increase the radii of
convergence, the global and local equation Eq (17)-(18) are subjected to regularization. The
details of the method can be found in [ ], and here we proceed to write down the final, axially
averaged and regularized model with its boundary and initial conditions –
(19)
is finite at (20)
. (21)
and, (22)

21. 21
CHAPTER 4
STEADY STATE
ANALYSIS

22. 22
STEADY STATE ANALYSIS
4.1 Bifurcation Diagrams
The steady state form of the regularized model without temporal and spatial gradients represents
the un-patterned or the base state. It is given by,
(23)
where, the steady state reaction rates are given by, (24)
Writing down, Eq. (16) for A and B, and adding them gives the steady state invariance,
(25)
Using, this invariance relation, we obtain a non-linear equation describing the base state of
species B.
(26)
This relation is used to construct steady state bifurcation diagrams of the species conversion with
Damkohler number (Da) as a parameter. The construction of the bifurcation diagrams employs
the arc-length method, which is presented in a later section.
We plot the steady state
conversion of A, vs. Da,
for different values of with
. The S-shaped
branch of the curve denotes
the unstable region, where
multiple steady states exist.
Hence, this is the region
suitable for pattern formation.
Figure 4 – Bifircation diagrams with parameter Pe

23. 23
Figure (4) shows that the region of multiplicity increases with decrease with increased axial
mixing.
In figure 5 we plot the
bifurcation diagram with the
ratio of inlet concentrations
as parameter. We find that
the region of multiplicity
decreases with increasing
.This is to be expected, since
for higher inlet
concentrations of B, the
autocatalytic effect is
enhanced, leading to rapid
consumption of A. Too less A is
left in the system to produce
patterns, and the system attains instantaneous steady state. Figure 5 shows that for a high value
of = 0.1, the system almost jumps from the extinguished branch to the ignited branch without
going through the unstable branch. The region of multiplicity is more for lower .
4.2 Hysteresis locii
The hysteresis locii present the region of multiple solutions in parameter space. In our case, we
choose to work in the parameter space spanned by Da and Pe. For constant , the values of Da
are plotted against 1/Pe . The hysteresis locii marks the region of multiplicity and hence also
demarcates the region favourable to pattern formation. Clearly, therefore, the boundary of such a
region must be those points in parameter space that correspond to the limit points on the S-
shaped bifurcation diagrams. This hints that the hysteresis locii region should be given by
solving in conjunction,
Figure 5 – Bifurcation diagrams with parameter

24. 24
and (27)
where f is given by Eq. (26)
The expressions for the limit point concentrations are obtained from,
which gives the equation for the hysteresis locii as –
(28)
Solving this gives,
(29)
where, .
The values of the limit point concentrations obtained in this way are plugged back into Eq. (26)
to give the generalised equation for hysteresis locii which is solved numerically to generate the
locii. As shown in figure (4), the locii is plotted in Da, Pe space for constant values of . The
region between the two limbs corresponds to the unstable middle branch of the bifurcation
diagram and hence represents the region for
pattern formation.
4.3 Linear Stability Analysis
The entire goal of obtaining a patterned state is to find out how a base state loses its stability
when subjected to transverse perturbations. Essentially, we need to determine the values of our
model parameters, Da and p, which are conducive to sustaining a perturbation. To achieve this,
we perturb the 2-D regularized model slightly, and carry out a linear stability analysis to find out
Figure 6 – Hysteresis Locii for =0.05, showing stable
and unstable regions

25. 25
if the perturbation is sustained, thus leading to pattern formation, or if it dies out. Since, the
underlying system geometry is a cylindrical co-ordinate system, we choose the cylindrical
eigenfunctions., viz. the different modes of the Bessel function in the radial direction as a
suitable perturbation. We introduce transverse perturbations of the form—
(30)
where, m and n are azimuthal and radial mode numbers respectively, is the coefficient of
temporal variation, that determines growth or decay, and is the axial eigenfunction, which in
this case is a constant, since the model is axially averaged. is the nth
non-trivial solution of
the equation that is obtained by subjecting the perturbation to the Neumann boundary condition
of the model i.e.
(31)
The values of obtained for different mode numbers have been tabulated here as –
We write the perturbed concentration as and plug it into Eq. (19). Further, we
note, that and we linearize the reaction term by doing a Taylor Expansion
around the base state to produce a system of coupled equations in terms of the constants ,
which can be written in matrix form as –
(32)
We wish to examine cases of neutral stability, i.e. the margin at which the system just crosses the
regimes of stability and consequently into a regime where spatial segregation starts happening,
Table 1 – Different values of for different m and n

26. 26
leading to the patterned state. So, we set . Now, for, Eq (22), to have non-trivial solutions,
we must have
(33)
This gives the desired criteria for Neutral Stability, as
(34)
The values of Da and can be obtained from the bifurcation diagrams presented in the
previous section. As discussed earlier, the bifurcation plots show the region of multiple steady
states, distinctly highlighting the values of Da for which the solution belongs to an extinguished
branch or an ignited (unnstable) branch. For the values of Da that belong to the unstable branch,
the neutral stability plots presented below, help us determine transverse Peclet number p, for
which the system will be unstable. Thus, to summarize, the operating parameter space of the 2-D
regularized model is
given by the pair (Da,
p).
Figure 7 presents the
neutral stability
boundary for 3 different
eigen modes. The
minimum value of the
transverse peclet
number in this stability
curve denotes the value
above which pattern
formation will occur.
Figure 7 – Neutral Stability curves for different eigenmodes

27. 27
The trends of the boundaries show that higher eigenmodes have a higher critical transverse peclet
number i.e. they will lead to patterns only under severe mixing limitations.
4.4 The Arc-Length Method
The bifurcation diagrams were generated with Arc-Length Methods, which are very common
schemes used in numerical continuation. The steady state bifurcation equation Eq. (19), is a
cubic in . It can always be solved by taking different values of the parameter Da, and then
applying any numerical scheme to solve the cubic for each parameter value. However, this is not
only computationally expensive, but also often, misses out important bifurcation points such as
folds, saddles and limit points. So, we use the Arc-Length method or its discrete counterpart, the
Psuedo Arc Length Method.
Consider a steady state solution in a variable-parameter space, .
For simplicity we consider scalars i.e. 1-variable, and 1-parameter. Suppose we know the
solution corresponding to the parameter value . Then, continuation essentially means
finding the solution at some later value of the parameter. To this end we make use of the arc-
length in variable-parameter space, and parameterize the curve in terms of this arc-length s.
(34)
We can immediately write,
(36)
Further, differentiating the curve at the point we have,
(37)
In the Arc-Length Method, Eqs (36) and (37) are solved as a set of nonlinear coupled ODEs and
the solution vector represents the bifurcation diagram, when x(s) is plotted vs.
p(s). In the Psuedo-Arc Length Method, we combine Eqs (36) and (37) to obtain,
(38)

28. 28
We start with a guess value of x and p, following which and are calculated using Eqs. (38)
and (36), respectively, in that order. Then x(s), is updated in successive arc-length iterations
using a Newton-Raphson progression –
(39)
In our case, we had, , and .

29. 29
CHAPTER 5
DYNAMIC
SIMULATION OF
PATTERNED STATES

30. 30
DYNAMIC SIMULATION OF PATTERNED STATES
In this section, we present spatio-temporal evolution of the base state into patterned states. To do
this, we carry out the steady state analyses as described above, which help us to obtain system
parameters favorable to pattern formation. The entire scheme, can be listed stepwise as –
 Obtain inlet conditions and perturbation mode numbers m,n as input.
 Generate the bifurcation plot for given , and select appropriate Da, belonging to the
unstable trajectory of the bifurcation diagram.
 Generate the subsequent Neutral Stability curve, and corresponding to the Da chosen in
the previous step, note the minimum p required for stability. Choose a p higher than this
critical value.
 With the suitable (p, Da) pair, provide around 3-12% perturbation to the initial base state
fr the corresponding Da. The perturbation corresponds to the mode numbers m and n.
Note that the perturbation should be kept within 12% , otherwise, the effect of the
perturbation will become significant and the patterns can no longer be called transport
limited.
 Solve the 2-D regularized model, Eqs. (19)-(21), using this perturbation as an initial
condition. The solution is advanced in time, and the temporal variation is observed.
5.1 Simulation Algorithm
The solution of Eqs (19)-(21) is a routine CFD problem. However, it is still instructive to discuss
the solution methodology followed in this project. The model simulations were carried out in
MATLAB, using its inbuilt 1-D partial differential equation solver PDEPE( ). Since, the
laplacian in Eq. (19) is two- dimensional, we proceed by discretizing the equation in the
azimuthal co-ordinate and solve the resulting set of coupled equation for different values of .
Post discretisation, the system looks like –
(40)

31. 31
where . When Eq (29) is written out for both species and for all the different
points of the grid, we get a discretised equation of the form,
(41)
where, is a vertical collocation of two triadiagonal matrices and is written as .
Eq. (42)
In this form, the system can be directly coded up using the 1-D PDE solver routines of
MATLAB.
5.2 Results and Discussions
Here, we present a few snapshots of temporal evolution of patterns, under different operating
conditions. The simulations were performed with an initial perturbation of 8% to the base state.
Table 2: The three basic patterns

32. 32
Table 2 shows the time evolution of the three basic perturbation modes that were used to excite
the system. These are the
 m=0, n=1: also called Target
 m=1, n=1: also called Band
 m=2, n=1: also called Antiphase
These modes are important, because all higher mode perturbations can be constructed by suitable
combinations of two or more of these modes
Table 3 presents a snapshot of the temporal evolution for the band mode for different values of
= 0.02, 0.05 and 0.01 keeping the other parameters fixed at and . We observe
Table 3: Patterns for different with m=1,n=1, Pe =1, p=100

33. 33
that for higher the pattern takes lesser time to decay. This is to be expected, since the region of
multiplicity (and consequently the regime favorable to pattern formation) in the bifurcation
diagram in Figure 5, decreases with an increased .
Table 4 presents the parametric dependence with axial Peclet number, Pe with other parameters
fixed at =0.02, m=1, n=1, p=100. The table reflects that with increase in axial mixing
limitations i.e. increase in Pe, the patterns decay slower. However, the effect of axial mixing is
not very effective in controlling the washout. This is to be expected in this analysis, since we are
working with a axially averaged model, which qualitatively captures the effect of Pe but may not
be able to predict the exact quantitative dependence.
Table 4: Patterns for different Pe with m=1,n=1, p=100

34. 34
Table 5 reflects the effect of the transverse Peclet number p. It is clear that the nature of the
effect is similar to that of the axial Peclet but, however, is much more pronounced. This is seen
in the large differences in decay times for different cases. The transverse Peclet is directly
embedded into the final low dimensional model that is solved and so Table 5 gives not only the
qualitative version of its effect but also captures the quantitative nature accurately. Washout is
quickest at p =10, which is the critical value for the band mode presented in above.
The effect of Damkohler number (Da) is similar to the effect of transverse Peclet. We shall
explain why this is so and hence shall not present an explicit table showing the dependence. Let
us examine what happens to the average concentration of a patterned state with time.
Table 5: Patterns for different p with m=1,n=1, Pe=1

35. 35
Figure 8 presents a comparison between traversals of the bifurcation diagrams with time. With
time, the average concentration of a patterned state,
is obtained. Then the average conversion at that time is mapped to the nearest point on the
bifurcation diagram for the given set of parameters. This enables us to find out how the patterned
state traverses the unstable middle branch of the bifurcation diagram from the extinguished state
to the ignited state (in terms of conversion). It is clearly seen from the comparison presented in
the above figure, that the lower the Damkohler number, the higher is the base state on the middle
branch. It is found that the perturbation kicks it up even higher. The overall effect is a higher
starting base state conversion for a lower Da, which in turn implies a lesser remaining path to
travel and hence a quicker decay time, which occurs as the conversion enters the ignited state.
So, the effect of Da will be similar to that of p. Lower the Da, lower is the reaction time-scale
and faster the depletion of species A, making the patterns to decay faster.
Figure 8 – Traversal of the bifurcation diagram with time

36. 36
CHAPTER 6
STABILITY OF
PATTERNED STATES

37. 37
STABILITY OF PATTERNED STATES
6.1 The notion of stability
So far, we have determined how a base state loses stability when subjected to transverse
perturbations and goes into a patterned state, which is characterized by intense segregation. In
this section, we shall go one step further and discuss methods of studying the stability of the
patterns formed. Before we begin discussing how to develop criterion for stability, it is important
to be absolutely clear about the notion of stability of a dynamically evolving pattern. In the usual
sense, stability can be always measured as the ability to return to its original form when
subjected to small perturbations. By this definition, if we take a snapshot of a particular patterned
state at a particular time point during its evolution as our starting pattern and perturb it slightly,
we will call it stable if the perturbations die down in time and it returns to its original state. We
must realize however, that since this state is dynamically evolving, it will have passed into
another state under conditions of zero perturbation too. Therefore, when perturbed, it will never
return to the original state. Instead it will transform into another patterned state which will then
subsequently die out gradually due to continuous transverse mixing.
Thus, stability for dynamic patterns requires a redefinition. In this project, we define stability as
the persistence of the patterned state (allowing for very slight deviations) for at least 3 to 4 time
iterations before it starts changing or washing away. The definition is extremely simple and
intuitive and must be converted into a proper mathematical or computational criterion.
6.2 Approaches of stability analysis of dynamic states
Here we shall discuss two basic methods of stability analysis as applied to the dynamic patterned
states in our present problem.
a) Neutral Stability Analysis
The linear stability analysis introduced earlier in section 4.3 is mostly applied to steady
states. However, it can be extended to this case. A computation of neutral stability criteria of a
patterned state can be easily shown to result in Eq. (34). Thus, we can pick up a pattern of our
choice and perturb it using a Bessel eigenmode (preferably the same that was used to excite this

38. 38
pattern) and compute the minimum transverse Peclet number (p) required to sustain this pattern
for a given value of Damkohler number (Da). This needs to be done for several sets of (p, Da)
and for a given axial Peclet number (Pe), this data will help generate a plot of p vs Da will
represent the stability boundary of the particular patterned state. Further to this, we will need to
perform a very extensive search in the (p, Da) space favourable to pattern formation to come up
with values of p and Da such that all of them produce the same spatial segregation as the given
pattern when halted after the same no. of time iterations. The resulting curve thus generated
will be the locii of the pattern. When superimposed on the original neutral stability map for the
base state, we will have three maps –
A steady locii with minimum at (say), a critical stability locii for the pattern with minimum at
(say), and the locii of the pattern
at the particular time iteration with
minimum at p(t) say. A pattern will
be stable when –
 to prevent washout
 t  to prevent
washout
  to sustain the
given pattern at (p(t), Da(t))
This premise is however, not without
flaw, since linear stability analysis, as mentioned previously is strictly speaking only applicable
for static states and further it is very rare if not completely improbable, that once perturbed a
dynamic pattern state will return to its original form.
b) Bifurcation Analysis
The approach that we propose to follow here is the bifurcation analysis of dynamic states. We
pick up a particular pattern and compute its bifurcation map and limit points. It must be made
clear that since these limit points are patterned states themselves, being functions of they
cannot be plotted on a S-shaped curve explicitly, as done in figure 2. Once these limit points are
Figure 9 – Neutral Stability analysis for dynamic states

39. 39
obtained, we plug them back into Eqs. (19)-(21) and obtain an expression for axial Peclet
number Pe, for the given Da. This points towards a hysteresis locus approach. For a given Da,
we can obtain Pe(t) as a function of time as the selected state evolves. Although no such explicit
bifurcation diagrams can be constructed, we present a schematic to explain what happens next.
Concentrate on figure (). If for a
given value of Da, the system
passes through only one limit point,
then it goes into the middle branch
2, which means it will come down
to the original dynamic state in
some time. However, if the system
passes through 2 limit points, it
crosses middle branch 2 and reaches
the excited state which an altogether
different dynamic state. Thus
 1 (or equivalently odd number of ) limit point(s)  stable pattern
 2(or 0 or even number of) limit points unstable pattern.
It is easy to see that the number of limit points is same as the number of solutions of Pe(t) at a
given t for a given value of Da. Thus generation of the hysteresis locus is indirectly a way to
predict whether the state of the pattern, if left to itself will be be stable or unstable. But before we
understand exactly how to go about it, it is necessary to formalize ourselves with some necessary
mathematics.
6.3 Frechet Derivatives and their computation
The Frechet Derivative is a derivative defined on Banach spaces that is commonly used to
generalize the derivative of a real valued function of a single variable to the case of a vector
valued functions of multiple variables. Let V and W be Banach spaces, and be an open
subset of V. A function is called Frechet differentiable at , if there exists a
bounded linear operator such that,
(42)
Figure 10 – Bifurcation analysis of dynamic state

40. 40
If this limit exists, we write and call it the Frechet Derivative of f at x. The Frechet
Derivative in finite dimensional space is the usual derivative. In particular it is represented in co-
ordinates by the Jacobian matrix. So for a vector function where, x is a vector of variables,
the Frechet Derivative Operator would be
, where each entry of J(x) is (43)
It should be kept in mind that is only an operator and. The value of the Frechet Derivative
at a point in the domain is only obtained when operates on a variable say v,
(44)
6.4 The 2-D Finite Fourier Transform in Plane Polar Co-ordinates
In this section, we shall develop a few workable notions of the 2-D Finite Fourier transform on a
circle. We shall not go into proving the claims as the reader is referred to [ ] for a more holistic
treatment of the principles. The 2-D Fourier transform on a rectangular domain can be easily
written as –
(45)
And the inverse Fourier transform is given by,
(46)
To extend these ideas to the plane polar co-ordinates , we note that any functional
dependence on these two can be decomposed into a Fourier series in the azimuth with
coefficients being functions of radial co-ordinates only.
where, (47)
Thus, when the function is split up into its radial and azimuthal components, its Fourier
transform can be treated as a series too, with radial and azimuthal parts which share a relation
similar to (47). It can be further shown that the radial part of the transform are solutions
of the Sturm-Liouville problem in plane polar co-ordinates, (we shall not go
into the proof here), which is why they can be given by the Hankel transform of order n of the
radial part of the original function, i.e.

41. 41
where, (48)
and, as discussed earlier, it can be shown that
(49)
With this understanding, we shall go ahead and directly write the transform and its inverse. The
complete 2-D forward Fourier transform in plane polar co-ordinates is given by,
(50)
For a finite [0, 1] domain (as in our case) the upper limits tending to will all become 1.
Further, for a finite domain, the inverse transform is given as —
(51)
An important property of the 2-D transform that is elementary to derive is stated here without
proof.
Property:
The Fourier transform of the Laplacian in 2-dimensions results in an eigenvalue problem in the
transform domain
Thus (52)
6.5 Stability maps and their development for a few test cases
With the necessary mathematical preliminaries now, we can carry out the bifurcation analysis of
the dynamic states. Consider a dynamic patterned state where i= A,B. The governing
equation is given by,
(53)

42. 42
To obtain the limit points, one must differentiate the function as was done in Eq (27), section 4.1.
However, what we have here is vector function and hence, we perform the Frechet Derivative
about the point . The equation of the limit points thus comes out to be,
(54)
where, . The systems of equations (53)-(54), must be solved together to find for
a given Da. That way we will be find Pe and perform the stability analysis by counting the
number of solutions. The rest of this section will be devoted to solving this system in an efficient
manner.
Subtracting the first entries of F and , we can write,
(55)
which is algebraic in nature and is hence solved in conjunction with .
The limit points are solved by exploiting the invariance properties of and using the 2-D
finite Fourier transform. Adding the two columns of we have the invariant form
(56)
where . Eq. (56) is solved by using the property of the transform stated in Eq. (52).
If be the transform, then the equation in transform domain is
(57)
the solution to which is (58)
This is plugged into the 1st
entry of and the resulting equation is subsequently
transformed. If be the 2-D transform, then in the transform
domain we have the inhomogenous equation,
(59)
The solutions to Eq (59) and correspondingly the transform for can be directly written after
solving as –

43. 43
(60)
where
(61)
Note that due to the presence of and , the transforms themselves are spatio-temporally
varying too. The transforms are inverted to obtain the limit points. The inversion is done as
explained in Eq. 51. The limit point concentrations when substituted into the algebraic relation
(55) yields a nonlinear equation in ,
(62)
where
. Also,
(63)
(j=A ,B) and s are solutions of the characteristic equation (31).
Thus we see that solution of Eq (62), would give us which would help us in
understanding how various regions within the pattern in question loses or gains stability. If we
were to label locations with even number of solutions as 0 (unstable), and odd number of
solutions as 1 (stable), then a false color image of dark and light regions moving against each
other would reveal what we call the stability map of the corresponding pattern and also show the
evolution of the stability map. In fact, this is exactly what we are going to do.
The computational schematic for generating stability maps can thus be set down as –
 Pick up a patterned state . j=A,B.
 Obtain initial transforms , by numerical integration.
 Obtain the function as given by Eq (63) by considering appropriate number of modes and
eigenvalues for each mode.

44. 44
 Obtain the stability map at any required subsequent time point.
A computer simulation of an evolving stability map was carried out with 5 eigenmodes and 5
eigenvalues of each. A pattern was chosen and its state at every 10th
iteration was picked up.
Also states close to the washout were taken. For each of these states, a stability map was
generated for a time that is half the original time-step from the given time point of the state. This
gives us an idea of instability creeps into the pattern and evolves with time. The determination of
the number of roots of Eq (62) was done by checking the number of intersection points of the
curves,
. After a few runs it was noticed that regions that became unstable in a particular iteration
remained so afterwards. This led us to conjecture that instability progresses continually. This
assumption was explicitly plugged into the code which greatly reduced the run-time.
Here, we present a few test cases and examine the dependence of parameters on the stability
maps.

45. 45
We begin with a stability map for a pattern given by the parameters , Pe=1, Da=14 and
p=100. We plot the pattern and the stability map together. The areas in dark brown are regions of
stability (with an odd number of limit points) while the lighter areas are zones of instability. We
observe that instability creeps in and then finally leads to washout of the pattern. We notice that
while the physical manifestation of the washout is the movement of a band, the actual
spreading of decay in the geometry occurs in a radial inward fashion. This is an important
result because, the spread is different across different modes and different eigenvalues. We
conjecture that it might be possible to establish an 1 to 1 correlation between how a pattern
decays and how its stability map evolves in time. Because of our previous conjecture about the
continual spread of instability across a pattern, to establish whether a particular state is stable or
Table 6: Stability map for , Pe=1 , p=100, Da=14

46. 46
not, we need to check the stability maps of states that come immediately after it. In this table, the
states at t=5 and t=15 are relatively steady states due to the low relative change in the stability
maps of states that follow them. By this logic, therefore the state at t=20 is extremely unstable,
which can also be seen in the massive difference in the structure of the patterns at t=20 and t=25.
However, note that stability maps may have little difference between them at certain time
iterations because of non-formation of patterned states yet. Hence, the segregation is another
factor that must be analysed in determining stability.
Thus, the principal steps of stability analysis can be summarized as –
 Obtain the stability map as a function of time.
 To determine stability of a particular state, compare its maps and its segregations
 Greater the similarity of the stability map of the state with that of its next state/(s) and higher
the segregation value of the state, more is the stability of the state.
It should once again be realized that the stability predicted thus is only a relative measure which
is what it should be. Since all states will eventually wash out, there cannot be anything like an
absolute measure stability of a particular state.
We perform a parametric analysis of stability with axial Peclet number. We no longer show the
patterns themselves but only the stability maps.

47. 47
Table 6 shows that with an increase in axial Peclet number from 5 to 10, the washout occurs
slower because of severe axial mixing limitations. However, the asymptotic case at Pe =0 seems
to be different from the trend in the sense that the decay pattern seems to be no longer radially
inward as is expected for band patterns but rather exhibiting band like behavior.
It has been seen in the previous chapter that patterns with higher modes and higher eigenvalues
have a higher critical transverse Peclet number (p). They can be excited only at very p and once
excited, prove to considerably resistant to diffusion. Patterns with low mode numbers on the
other hand readily wash out. Thus, it is interesting to see if higher mode patterns are indeed more
stable than lower mode patterns.
Table 7: Stability map for , p=100 and Pe = 0,5 and 10

48. 48
Table 8: Stability Maps of symmetric eigenmodes for Pe=1, Da=10 and critical p
for the mode
Figure 11: Stability of modes expressed as % of stable node points

49. 49
Table 7 and Figure 10 properly summarize our find and establish that stability does indeed
increase with higher modes. We plotted the stability maps for each mode considering an
eigenvalue equal to the mode number. The results of Table 7 suggest at a glance that the number
of stable zones (dark zones) increase with the mode number. Further the nature of the maps
change little across time iterations as we go higher in the mode number. We further establish this
quantitatively as well. Selecting % of stable node points as a suitable metric, we plot the time
variation of this percentage and rather compare the time averaged value of this percentage for
different modes. The simulations reveal that stability increases till m=3 and then decreases
slightly. This hints towards the existence of two conflicting factors –
a) When a higher eigenmode is excited, it excites with it the lower modes too. This leads to
a broad range of modes which in turn result in superimposition of patterns with widely
differing length scales. This is brings in more asymmetry and consequently more
stability.
b) At higher eigenmodes segregation intensifies. This means that regions of very low and
very high concentrations can co-exist thus causing additional local driving force for
diffusion.
Evidently (a) is responsible for the initial upward trend in stability, while (b) is responsible for
the downward trend.
Finally, we demonstrate how to obtain the steady pattern for a given set of operating conditions
and a maximum run time. In this demonstration we consider the operating conditions to be same
as in Table 7. We obtain a plot of the two conflicting trends discussed above. Percentage of
stable node points and percentage segregation in a state are plotted as functions of time. For the
mode with m=5, n=5 the plot is presented.

50. 50
Clearly, the most stable pattern for this set of operating conditions is the one where the 2 curves
intersect. Thus, the required most stable pattern is formed at iteration # 4 or t =20. Using this
procedure, the most stable patterns corresponding to Table 7 are presented here.
.
Figure 12 – Stability and Segregation (conflicting trends) for m=5, n=5
Table 9: Stable patterns for various mode numbers

51. 51
CHAPTER 7
CONCLUSIONS AND
FUTURE WORK

52. 52
CONCLUSIONS AN FUTURE WORK
7.1 Conclusions
Our simulations show that symmetric and asymmetric patterns emerge from the unstable middle
branch of the S-shaped bifurcation curve due to small perturbations to the uniform steady states,
and undergo a process of concentration segregation. A detailed parametric analysis shows the
variation of the patterns formed with different values of p, Pe, Da, and μ. We conclude that the
presence of significant transverse mixing limitations provides the necessary and sufficient
condition for pattern formation. Axial mixing limitations only provide supplementary effects by
making the patterns emerge faster and become more stable. Hence, it can be said that increased
transverse or axial mixing limitations (quantified by increasing p and Pe, respectively) result in
patterns that are formed sooner and attain stability faster. We establish the concept of stability of
a patterned state and examine the stability map of a few test cases. Further we study the temporal
variation of such stability maps and also lay down rules for determining stability of a pattern
given its starting operating parameters.
7.2 Future work
The most important step that must be taken is to formulate other metrics of stability of a dynamic
state, preferably metrics which do not vary in space across the pattern. Further, it remains to
study the properties of stability maps and also to see if stability maps of different patterns
starting at the same time overlap after several time iterations, and if so to extract physical
meaning from it. The change in the nature of decay of a pattern at zero Pe from the decay nature
at other Pe values is interesting and needs further investigation.

53. 53
CHAPTER 8
REFERENCES

54. 54
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